Electornic supplemnt:
A Seismologically Consistent Surface Rupture Length Model for
Unbounded and Width-Limited Events

Introduction

This study presents a new \(SRL\) model with physically defensible scaling for unbounded and width-limited ruptures.

Study Objectives:

• Consistent scaling with seismological theory
• Captures scaling for width-limited ruptures

Load Data

The empirical data that are used in model development include the FDHI (Sarmiento et al. 2021), Wells and Coppersmith (1994), and Baize et al. (2020) datasets. The surface rupture lengths (\(SRL\)) of the events in FDHI dataset are computed based on the Event Coordinate System (ECS), while the \(SRL\) in Wells and Coppersmith (1994) and Baize et al. (2020) are reported directly. Additionally, dynamic rupture simulations performed by SCEC are used to derive the rupture width model for the width-limited ruptures.

#read srl flatfiles
#FDHI
df_FDHI_srl_all <- read.csv(fname_FDHI_srl)

#FDHI processing
#create normal and reverse fault flags
df_FDHI_srl_all$FN <- pmax(-df_FDHI_srl_all$sof, 0)
df_FDHI_srl_all$FR <- pmax( df_FDHI_srl_all$sof, 0)
df_FDHI_srl_all$FD <- abs(  df_FDHI_srl_all$sof )
#style of faulting name
df_FDHI_srl_all$style2                         <- 'Strike-Slip'
df_FDHI_srl_all[df_FDHI_srl_all$FN>0,'style2'] <- 'Normal'
df_FDHI_srl_all[df_FDHI_srl_all$FR>0,'style2'] <- 'Reverse'
#fault width
df_FDHI_srl_all$seis_width <- df_FDHI_srl_all$seis_depth / sin(deg2rad(df_FDHI_srl_all$dip))

#SCEC
#read srl flatfiles
df_SCEC_srl_original  <- read.csv(fname_SCEC_srl_original)
df_SCEC_srl_low_width <- read.csv(fname_SCEC_srl_low_width)
df_SCEC_srl_const_sig <- read.csv(fname_SCEC_srl_const_sig)
df_SCEC_srl_sim_2     <- read.csv(fname_SCEC_srl_sim2)
df_SCEC_srl_sim_3     <- read.csv(fname_SCEC_srl_sim3)
df_SCEC_srl_sim_4     <- read.csv(fname_SCEC_srl_sim4)

#SCEC processing
#define seimogenic depths
df_SCEC_srl_original$seis_depth  <- 20
df_SCEC_srl_low_width$seis_depth <- 15 
df_SCEC_srl_const_sig$seis_depth <- 20
df_SCEC_srl_sim_2$seis_depth     <- 20
df_SCEC_srl_sim_3$seis_depth     <- 20 
df_SCEC_srl_sim_4$seis_depth     <- 20 
#merged data-frame of different simulations
df_SCEC_srl <- bind_rows( df_SCEC_srl_original, df_SCEC_srl_low_width, df_SCEC_srl_const_sig,
                          df_SCEC_srl_sim_2, df_SCEC_srl_sim_3, df_SCEC_srl_sim_4)
#identify width limited events
df_SCEC_srl$flag_wlim <- FALSE
df_SCEC_srl[df_SCEC_srl$wef >= df_SCEC_srl$seis_depth-2.5,'flag_wlim'] <- TRUE 
rm(list=c('df_SCEC_srl_original','df_SCEC_srl_low_width','df_SCEC_srl_const_sig',
          'df_SCEC_srl_sim_2', 'df_SCEC_srl_sim_3', 'df_SCEC_srl_sim_4'))

The maximum fault width of the empirical data was calculated based on the thickness of the seismogenic zone (\(D_{seis}\)) and fault dip angle (\(\theta\)). Table 1 summarizes \(D_{seis}\) based on the fault location and Table 2 summarizes \(\theta\) for different styles of faulting. \[ W_{lim} = \frac{D_{seis}}{\sin(\theta)} \] \(W_{lim}\) is the maximum width, a rupture can reach before it becomes width limited.

Table 1. Seismogenic Zone Thickness for Differnt Regions

Region	Seismogenic Depth (km)
California	15
Guatemala	15
New Zealand	15
Indonesia	15
Japan	17
Himalayan Region	30
Australia	40
Other	22

Table 2. Dip Angles for Differnt Styles of Faulting

Style of Fauling	Dip Angle (deg)
Strike-Slip	90
Normal	60
Reverse	45

Outliers

The following events were excluded from the regression dataset:

• Events less than \(\mathbf{M}~5\)
• 1892 Laguna Salada: it was occurred in 1892 but mapped in the 2010s. Magnitude is estimated.
• 1978 IzuOshima: surface is incomplete, rupture probably extended offshore to southeast; epicenter is offshore ~17 km
• 1993 Killari: it occurred in a stable continental region; probably a lot of deformation accommodated via folding/warping that was unmapped

#events for regression
df_FDHI_srl <- df_FDHI_srl_all
#remove outliers 
df_FDHI_srl  <- subset(df_FDHI_srl, !is.na(event_len) & flag_missing==0 & flag_multiple==0 )
#1892 Laguna Salada
df_FDHI_srl  <- subset(df_FDHI_srl, !(EQ_ID==40 & FLT_ID==1))
#1978 IzuOshima
df_FDHI_srl  <- subset(df_FDHI_srl, !(EQ_ID==62 & FLT_ID==1))
#1993 Killari
df_FDHI_srl  <- subset(df_FDHI_srl, !(EQ_ID==74 & FLT_ID==1))
#2010 Pisayambo
df_FDHI_srl <- subset(df_FDHI_srl,  !(EQ_ID==58 & FLT_ID==1))
# #1957 Gobi-Altai
# df_FDHI_srl <- subset(df_FDHI_srl,  !(EQ_ID==52 & FLT_ID==2))
#2018 Etna
df_FDHI_srl <- subset(df_FDHI_srl,  !(EQ_ID==6 & FLT_ID==3))


#remove mag less than 5
df_FDHI_srl  <- subset(df_FDHI_srl, magnitude>=5)

#summarize all outliers
df_FDHI_srl_out <- subset(df_FDHI_srl_all,!(rownames(df_FDHI_srl_all)%in%rownames(df_FDHI_srl)))

Summary datasets

Summary statistics of the empirical dataset are presented below:

## Number of Events per dataset:
##           Total: 122
##           FDHA: 64
##           W&C: 52
##           Baize: 6

## Number of Events per region:
##           Total: 122
##           California: 29
##           Guatemala: 1
##           New Zealand: 3
##           Indonesia: 1
##           Japan: 7
##           Himalayan Region: 12
##           Australia: 11
##           Other: 58

## Number of Events per Style of Faulting:
##           Strike-slip: 64
##           Normal: 26
##           Reverse: 32

## Magnitude Range:
##           maximum: 8.1
##           minimum: 5.0

## Rupture-Length Range:
##           maximum: 432.0
##           minimum: 1.1

Figure 1 shows the Magnitude - Surface rupture length (\(\mathbf{M} - SRL\)) distribution of the empirical dataset.

Figure 1. Magnitude - SRL distribution of selected empirical data

Summary statistics of the SCEC simulations are presented below:

## Total number of simulations: 554

## Magnitude Range:
##           maximum: 8.2
##           minimum: 4.9

## Rupture-Length Range:
##           maximum: 655.2
##           minimum: 1.0

Figure 2 compares the \(\mathbf{M} - SRL\) distribution of empirical data and SCEC simulations. The two datasets show similar scaling supporting that the scaling relationships derived from the SCEC simulations are applicable to the empirical data.

Figure 2. Comparison of Magnitude - SRL distribution of empirical data and SCEC simulations

Model Development

Rupture Width Modeling Based on SCEC Simulations

The SCEC simulations were used to develop a Magnitude - Rupture width model for unbounded and width-limited ruptures. Two models were evaluated: (i) a linear model between \(W - \mathbf{M}\), hereafter denoted as model w0, and (ii) a model with a scaling break at \(W_{lim}\), hereafter denoted as model w1

The functional form for model_w0 is: \[\begin{equation} \log(W) = \mathbf(M) + \epsilon_W \end{equation}\]

The functional form for model_w1 is: \[\begin{equation} log(W) = \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) + \epsilon_W \end{equation}\]

## Regression formulas
form_model_w0 <- log10(wef) ~ 1 + magnitude
form_model_w1 <- log10(wef) ~ 1 + magnitude

## Regression
fit_model_w0 <- lm(form_model_w0, data=df_SCEC_srl)
fit_model_w1 <- lm(form_model_w1, data=subset(df_SCEC_srl, !flag_wlim))

The \(W - \mathbf{M}\) model summary is presented below. model w1 has a higher log likelihood and lower Akaike information criterion.

## Width Modeling
## 
##              Model0:
##      LogLik: 641.71
##      AIC: -1277.43
## 
##              Model1:
##      LogLik: 889.34
##      AIC: -1772.69

The best estimates of the coefficients of the two candidate models are summarized in Table 3.

Table 3. Rupture Width Model Coefficients

	Intercept	\(\mathbf{M}\) scaling	\(\sigma\)
model_w0	-0.7603617	0.2735708	0.0761189
model_w1	-1.1602315	0.3395897	0.0544508

Figure 3 compares the two candidate models against the SCEC simulations.

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

Figure 3. Comparison of candidate rupture width models against the SCEC simulations

The summary of the model w1 regression is provided below:

summary(fit_model_w1)

## 
## Call:
## lm(formula = form_model_w1, data = subset(df_SCEC_srl, !flag_wlim))
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.123926 -0.033305  0.000471  0.036243  0.172420 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -1.160232   0.032544  -35.65   <2e-16 ***
## magnitude    0.339590   0.005303   64.04   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.05445 on 328 degrees of freedom
## Multiple R-squared:  0.9259, Adjusted R-squared:  0.9257 
## F-statistic:  4101 on 1 and 328 DF,  p-value: < 2.2e-16

Figure 4 shows the scaling of the preferred model (model w1) against the SCEC data.

## Warning: The `size` argument of `element_rect()` is deprecated as of ggplot2 3.4.0.
## ℹ Please use the `linewidth` argument instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

Figure 4. Preferred model for rupture width scaling

Figure 5 shows the residuals of the preferred model versus magnitude.

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 5. Resisuals of Preferred Rupture Width Model for SCEC simulations

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

SRL Model Derivation

This section presents the deviation of the \(SRL\sim\mathbf{M}\) functional form. The Seismic Moment (\(M_0\)) is defined as:

\[\begin{equation} M_0 = \mu~\bar{D}~L~W \end{equation}\]

where \(\mu\) is the stiffness of the crust, \(\bar{D}\) is the average slip on the rupture, \(L\) is the length of the rupture, and \(W\) is width of the rupture. Taking the log of the previous equation and substituting the definition of the Moment Magnitude (\(\mathbf{M} = \frac{2}{3}\log(M_0) - 10.7\)) we get:

\[\begin{equation} \mathbf{M} = \frac{2}{3}\log(\mu) + \frac{2}{3}\log(\bar{D}) + \frac{2}{3}\log(L) + \frac{2}{3}\log(W) - 10.7 \end{equation}\]

Since stiffness of the crust (\(\mu\)) is constant, the previous equation simplifies to:

\[\begin{equation} \mathbf{M} = \frac{2}{3}\log(\bar{D}) + \frac{2}{3}\log(L) + \frac{2}{3}\log(W) + c \end{equation}\]

where \(c\) is used to denote any arbitrary constant.

Three commonly used scaling models for the average displacements found in the literature include the \(\sqrt{A}\) model, hereafter called model 1, the \(W\) model, hereafter called model 2, and the \(L\) model, hereafter called model 3 (Scholz, 1982; Romanowicz, 1992; Romanowicz and Rundle, 1993; Romanowicz, 1994; Scholz, 1994; Bodin and Brune, 1996; Pegler and Das, 1996; Scholz, 1997, 1998; Hanks and Bakun, 2002; Leonard, 2010).

For model 1, the average displacement is assumed to scale linearly with the square root of the rupture area:

\[\begin{equation} \bar{D} = c \sqrt{A} = c \sqrt{L~W} \end{equation}\]

For model 2, the average displacement is assumed to scale linearly with the width of the rupture:

\[\begin{equation} \bar{D} = c W \end{equation}\]

while for model 3, the average displacement is assumed to scale linearly with the length of the rupture:

\[\begin{equation} \bar{D} = c L \end{equation}\]

Using model 1 the \(\mathbf{M}\) scaling relationship becomes: \[\begin{equation} \mathbf{M} = \log(L) + \log(W) + c \end{equation}\] with model 2 the \(\mathbf{M}\) scaling relationship becomes: \[\begin{equation} \mathbf{M} = \frac{2}{3}\log(L) + \frac{4}{3}\log(W) + c \end{equation}\] and with model 3 the \(\mathbf{M}\) scaling relationship becomes: \[\begin{equation} \mathbf{M} = \frac{4}{3}\log(L) + \frac{2}{3}\log(W) + c \end{equation}\]

Solving for \(\log(L)\) in previous equations model 1 becomes: \[\begin{equation} \log(L) = \mathbf{M} - \log(W) + c \end{equation}\] model 2 becomes: \[\begin{equation} \log(L) = \frac{3}{2} \mathbf{M} - 2 \log(W) + c \end{equation}\] and model 3 becomes: \[\begin{equation} \log(L) = \frac{3}{4} \mathbf{M} - \frac{1}{2} \log(W) + c \end{equation}\]

An additional two candidate models are developed which follow \(\sqrt{A}\) scaling for unbounded ruptures and transition to \(L\) and \(W\) scaling, respectively, for width-limited events. The two models are referred to as model 2' and model 3' and their functional forms are provided next. The model 2' functional form is: \[\begin{equation} \log(L) = \begin{cases} \mathbf{M} - \log(W) + c & \mathbf{M} \le \mathbf{M}_{lim} \\ \frac{3}{2} \mathbf{M} - 2 \log(W) + c_2' + c & \mathbf{M} > \mathbf{M}_{lim} \end{cases} \end{equation}\] while, the model 3' functional form is: \[\begin{equation} \log(L) = \begin{cases} \mathbf{M} - \log(W) + c & \mathbf{M} \le \mathbf{M}_{lim} \\ \frac{3}{4} \mathbf{M} - \frac{1}{2} \log(W) + c_3' + c & \mathbf{M} > \mathbf{M}_{lim} \end{cases} \end{equation}\] \(c_2'\) and \(c_3'\) are added in order to have a continuous transition between the two branches. The transition from unbounded to width-limited events occurs at \(\mathbf{M}_{lim}\), \(W_{lim}\) coordinate pair. Solving for \(c_2'\) and \(c_3'\) at the transition point we get:

\[\begin{equation} c_2' = -\frac{1}{2} \mathbf{M}_{lim} + \log(W_{lim}) \end{equation}\] \[\begin{equation} c_3' = \frac{1}{4} \mathbf{M}_{lim} - \frac{1}{2} \log(W_{lim}) \end{equation}\] With the estimated coefficients, the previous equations can be simplified. model 2' becomes: \[\begin{equation} \log(L) = \frac{3}{2} \mathbf{M} - \frac{1}{2} \min(\mathbf{M},\mathbf{M}_{lim}) - \log(W) + c \end{equation}\] and model 3' becomes: \[\begin{equation} \log(L) = \frac{3}{4} \mathbf{M} + \frac{1}{4} \min(\mathbf{M},\mathbf{M}_{lim}) - \log(W) + c \end{equation}\]

Substituting the model w1 from the SCEC rupture simulations to the model 1 functional form we get: \[\begin{equation} \log(L) = \mathbf{M} - \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) + c \end{equation}\] for model 2 we get: \[\begin{equation} \log(L) = \frac{3}{2} \mathbf{M} - 2 min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) + c \end{equation}\] and for model 3 we get: \[\begin{equation} \log(L) = \frac{3}{4} \mathbf{M} - \frac{1}{2} min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) + c \end{equation}\]

Similarly, for model 2' we get: \[\begin{equation} \log(L) = \frac{3}{2} \mathbf{M} - \frac{1}{2} \min(\mathbf{M},\mathbf{M}_{lim}) - min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) + c \end{equation}\] and for model 3' we get: \[\begin{equation} \log(L) = \frac{3}{4} \mathbf{M} + \frac{1}{4} \min(\mathbf{M},\mathbf{M}_{lim}) - min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) + c \end{equation}\]

Assuming the surface to sub-surface rupture length scaling is of the form \(\log(SRL) = c_1 \log(L) + c\), substituting it to the previous equations, condensing the constants, and solving for \(\log(SRL)\), for model 1 we get: \[\begin{equation} \log(SRL) = c_1 \left( \mathbf{M} - \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) \right) + c_2 = c_1 X_1 + c_2 \end{equation}\] for model 2 we get: \[\begin{equation} \log(SRL) = c_1 \left( \frac{3}{2} \mathbf{M} - 2 \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) \right) + c_2 = c_1 X_2 + c_2 \end{equation}\] for model 3 we get: \[\begin{equation} \log(SRL) = c_1 \left( \frac{3}{4} \mathbf{M} - \frac{1}{2} \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) \right) + c_2 = c_1 X_3 + c_2 \end{equation}\] for model 2' we get: \[\begin{equation} \log(SRL) = c_1 \left( \frac{3}{2} \mathbf{M} - \frac{1}{2} \min(\mathbf{M},\mathbf{M}_{lim}) - \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) \right) + c_2 = c_1 X_2' + c_2 \end{equation}\] and for model 3' we get: \[\begin{equation} \log(SRL) = c_1 \left( \frac{3}{4} \mathbf{M} + \frac{1}{4} \min(\mathbf{M},\mathbf{M}_{lim}) - \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})) \right) + c_2 = c_1 X_3' + c_2 \end{equation}\]

The previous equations can be expressed as linear models with, \(X_1 = \mathbf{M} - \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})\), \(X_2 = \frac{3}{2} \mathbf{M} - 2 \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})\), \(X_3 = \frac{3}{4} \mathbf{M} - \frac{1}{2} \min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim})\), \(X_2' = \frac{3}{2} \mathbf{M} - \frac{1}{2} \min(\mathbf{M},\mathbf{M}_{lim}) - min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim}))\), and \(X_3' = \frac{3}{4} \mathbf{M} + \frac{1}{4} \min(\mathbf{M},\mathbf{M}_{lim}) - min(\beta_1 + \beta_2 \mathbf{M}, \log(W_{lim}))\).

The effect of the style of faulting is also considered through the use of additive terms to account for differences in the static stress drop. The static stress drop is defined as: \[\begin{equation} \Delta \sigma = C \mu \frac{\bar{D}}{L_C} \end{equation}\] thus, in log scale, a shift in the stress drop will be accompanied by a linear shift in the average displacement (\(\log(\bar{D}) = \log(\Delta \sigma) + const\)), which, in turn, will shift the intercept of the \(SLR \sim \mathbf{M}\) models
Two modeling approaches were evaluated: (i) a common additive term for dip-slip events (\(F_D\)), and (ii) separate additive terms for Normal and Reverse faults (\(F_N\) and \(F_R\)). The style of faulting dependence introduced through \(F_D\), \(F_N\), and \(F_R\) acts in addition to the style of faulting dependence introduced by the different dip angles between the different styles of faulting.

The final regression relationship for model 1 is: \[\begin{equation} \log(SRL) = c_1 X_1 + c_2 F_D + c_3 \end{equation}\] the final regression relationship for model 2 is: \[\begin{equation} \log(SRL) = c_1 X_2 + c_2 F_D + c_3 \end{equation}\] the final regression relationship for model 3 is: \[\begin{equation} \log(SRL) = c_1 X_3 + c_2 F_D + c_3 \end{equation}\] the final regression relationship for model 2' is: \[\begin{equation} \log(SRL) = c_1 X'_2 + c_2 F_D + c_3 \end{equation}\] the final regression relationship for model 3' is: \[\begin{equation} \log(SRL) = c_1 X'_3 + c_2 F_D + c_3 \end{equation}\] and the final regression for separate additive terms with \(D \sim \sqrt{A}\) scaling, hereafter denoted as model 1alt is: \[\begin{equation} \log(SRL) = c_1 X_1 + c_2 F_N + c_3 F_R + c_4 \end{equation}\]

Lastly a Wells and Coppersmith type model was also evaluated, hereafter referred as model 0, to investigate the impact of width-limited ruptures on current state-of-practice models. \[\begin{equation} \log(SRL) = c_1 \mathbf{M} + c_2 F_D + c_3 \end{equation}\]

The coefficients of the previous equations are estimated through a maximum likelihood regression.

Surface Rupture Length Modeling Based on Empirical Data

The seven candidate models evaluated in predicting \(SRL\) as a function of \(\mathbf{M}\) are summarized in Table 4.

Table 4. Candiate Regression Models

Name	Functional Form	Description
model 0	\(\log(SLR)\sim 1 + \mathbf{M} + F_D\)	W&C 94 type model
model 1	\(\log(SLR)\sim1 + X_1 + F_D\)	\(\bar{D} \sim \sqrt{A}\) type model with \(F_D\) addative term
model 2	\(\log(SLR)\sim 1+X_2 + F_D\)	\(\bar{D} \sim {W}\) type model with \(F_D\) addative term
model 3	\(\log(SLR)\sim 1+X_3 + F_D\)	\(\bar{D} \sim {L}\) type model with \(F_D\) addative term
model 2’	\(\log(SLR)\sim 1+X_{2'} + F_D\)	\(\bar{D} \sim \sqrt{A},{W}\) type model with \(F_D\) addative term
model 3’	\(\log(SLR)\sim 1+X_{3'} + F_D\)	\(\bar{D} \sim \sqrt{A},{L}\) type model with \(F_D\) addative term
model 1 alt	\(\log(SLR)\sim1 + X_1 + F_N+F_R\)	\(\bar{D} \sim \sqrt{A}\) type model with \(F_N\) & \(F_R\) addative terms

The next part shows the definition of the regression functional forms and model fit.

# Rupture Width Modeling
#rupture width coefficients
c1_rw <- df_SCEC_model_width_mag_coeffs['model_w1','intrcp']
c2_rw <- df_SCEC_model_width_mag_coeffs['model_w1','M']
#compute rupture widths 
df_FDHI_srl$rup_width_unbnd <- 10^(c1_rw + c2_rw*df_FDHI_srl$magnitude)
df_FDHI_srl$rup_width <- pmin(df_FDHI_srl$rup_width_unbnd, df_FDHI_srl$seis_width)

#transitioning magnitude
df_FDHI_srl$magnitude_lim <- 1/c2_rw * (log10(df_FDHI_srl$seis_width) - c1_rw )

#identify width limited events
df_FDHI_srl$flag_wlim <- FALSE
df_FDHI_srl[df_FDHI_srl$rup_width >= df_FDHI_srl$seis_width,'flag_wlim'] <- TRUE 

# SRL Modeling
#scaling parameters
#simple models
df_FDHI_srl$x1 <-     df_FDHI_srl$magnitude -   log10(df_FDHI_srl$rup_width)
df_FDHI_srl$x2 <- 3/2*df_FDHI_srl$magnitude - 2*log10(df_FDHI_srl$rup_width)
df_FDHI_srl$x3 <- 3/4*df_FDHI_srl$magnitude - 1/2*log10(df_FDHI_srl$rup_width)
#composite models
df_FDHI_srl$x2prime <- 3/2*df_FDHI_srl$magnitude - 1/2*pmin(df_FDHI_srl$magnitude,df_FDHI_srl$magnitude_lim) - log10(df_FDHI_srl$rup_width)
df_FDHI_srl$x3prime <- 3/4*df_FDHI_srl$magnitude + 1/4*pmin(df_FDHI_srl$magnitude,df_FDHI_srl$magnitude_lim) - log10(df_FDHI_srl$rup_width)

#standard deviation weights
width_scl <- 0.1
sig_scl   <- 2.
df_FDHI_srl$reg_wt <- ( 1 + (sig_scl-1) * sigmoid( 2.197225 * (log10(df_FDHI_srl$rup_width_unbnd)-log10(df_FDHI_srl$seis_width))/ width_scl ) )^2

## Regression formulas
form_model_srl0m     <- log10(event_len) ~ 1 + magnitude  + FD
form_model_srl1m     <- log10(event_len) ~ 1 + x1         + FD
form_model_srl2m     <- log10(event_len) ~ 1 + x2         + FD
form_model_srl3m     <- log10(event_len) ~ 1 + x3         + FD
form_model_srl2pm    <- log10(event_len) ~ 1 + x2prime    + FD
form_model_srl3pm    <- log10(event_len) ~ 1 + x3prime    + FD
form_model_srl1m_alt <- log10(event_len) ~ 1 + x1         + FN + FR

## Regression
fit_model_srl0m     <- lm(form_model_srl0m,     data=df_FDHI_srl)
fit_model_srl1m     <- lm(form_model_srl1m,     data=df_FDHI_srl, weights=reg_wt)
fit_model_srl2m     <- lm(form_model_srl2m,     data=df_FDHI_srl, weights=reg_wt)
fit_model_srl3m     <- lm(form_model_srl3m,     data=df_FDHI_srl, weights=reg_wt)
fit_model_srl2pm    <- lm(form_model_srl2pm,    data=df_FDHI_srl, weights=reg_wt)
fit_model_srl3pm    <- lm(form_model_srl3pm,    data=df_FDHI_srl, weights=reg_wt)
fit_model_srl1m_alt <- lm(form_model_srl1m_alt, data=df_FDHI_srl, weights=reg_wt)

The print out below summarizes the log-likelihood and Akaike Information Criterion (AIC) for the different candidate models, while Table 5 summarizes the estimated coefficients.

From a statistical perspective, the simple models (model 1 to model 3) and composite model 2' have similar good predictive performance, with model 1 being slightly better. The composite model 3' has poorer performance, while model 0 has the worst performance. model 0 does not consider the finite thickness of the seismogenic zone showcasing the limitations of a purely empirical model. The candidate models (model 1 to model 3') have a break in the magnitude scaling when the width of the rupture reaches the width of the seismogenic zone allowing them to better fit the data.

## SRL Modeling:
##          Model0:
##      LogLik: 6.06
##      AIC: -4.12
##  
##          Model1:
##      LogLik: 15.42
##      AIC: -22.85
## 
##          Model2:
##      LogLik: 14.15
##      AIC: -20.30
## 
##          Model3:
##      LogLik: 14.30
##      AIC: -20.60
## 
##          Model2':
##      LogLik: 8.91
##      AIC: -9.81
## 
##          Model3':
##      LogLik: 12.22
##      AIC: -16.44
## 
##          Model1 alt.:
##      LogLik: 15.43
##      AIC: -20.86

## Standard Deviations:
##  Model 1:
##      Empirical ratio for std of width limted to unbounded events is: 1.93
## 
##          Standard Deviations:
##  Model 2:
##      Empirical ratio for std of width limted to unbounded events is: 1.88
## 
##          Standard Deviations:
##  Model 3:
##      Empirical ratio for std of width limted to unbounded events is: 1.88
## 
##          Standard Deviations:
##  Model 2':
##      Empirical ratio for std of width limted to unbounded events is: 1.74
## 
##          Standard Deviations:
##  Model 3':
##      Empirical ratio for std of width limted to unbounded events is: 1.80
## 
##          Standard Deviations:
##  Model 1 alt.:
##      Empirical ratio for std of width limted to unbounded events is: 1.92

Table 5. SRL Model Coefficients

	Intercept	\(\mathbf{M}\) scaling	\(X_1\) scaling	\(X_2\) scaling	\(X_3\) scaling	\(X_{2'}\) scaling	\(X_{3'}\) scaling	\(F_D\)	\(F_N\)	\(F_R\)	\(\sigma\)
model_srl0	-2.968009	0.6628783	NA	NA	NA	NA	NA	-0.1836268	NA	NA	0.2331305
model_srl1	-3.878287	NA	0.9509412	NA	NA	NA	NA	-0.1517421	NA	NA	0.2494025
model_srl2	-4.179800	NA	NA	0.7152433	NA	NA	NA	-0.1390408	NA	NA	0.2520182
model_srl3	-3.604864	NA	NA	NA	1.131573	NA	NA	-0.1645242	NA	NA	0.2517162
model_srl2p	-3.097664	NA	NA	NA	NA	0.8044237	NA	-0.1291246	NA	NA	0.2630927
model_srl3p	-4.250663	NA	NA	NA	NA	NA	1.023459	-0.1758600	NA	NA	0.2560359
model_srl1_alt	-3.881661	NA	0.9515046	NA	NA	NA	NA	NA	-0.155196	-0.1480471	0.2504442

From seismological theory, model 1 to model 3' should have unit slopes with respect to the magnitude scaling terms (e.g. \(X_1\), \(X_2\), \(X_3\)), if the average displacement in the empirical data follows the assumed scaling law and the subsurface rupture length (\(L\)) was the response variable. With \(SRL\) as a response variable, a magnitude scaling term greater than one would imply that the surface to sub-surface rupture length ratio increases with magnitude while a magnitude scaling term less than one would imply that the surface to sub-surface rupture length ratio decreases with magnitude. Intuitively, a greater than one magnitude scaling term is expected, as for large events, a larger part of the rupture is expected to reach the surface.

Based on the derived slopes, model 1 and model 3' show a data-derived scaling consistent with seismological theory, with \(X_1\) scaling in model 1 being slightly less than unity, and \(X_3'\) scaling in model 3' being slightly above unity. Considering the statistical performance and seismological scaling of the candidate models, model 1 with a fixed magnitude scaling slope equal to unity is selected as a preferred model, hereafter called model1_fxd, and model 3' with fixed unit slope is denoted as an alternative valid model, hereafter called model3_fxd'. In the selection process, seismologically consistent scaling is weighted more heavily than statistical performance, due to the small size of the empirical data set and importance of physically plausible model extrapolation seismic hazard analyses.

## Preferred model 
# regression formula
form_model_srl1m_fxd  <- log10(event_len) ~ 1 + offset(x1) + FD
form_model_srl3pm_fxd <- log10(event_len) ~ 1 + offset(x3prime) + FD
# regression
fit_model_srl1m_fxd  <- lm(form_model_srl1m_fxd,  data=df_FDHI_srl, weights=reg_wt)
fit_model_srl3pm_fxd <- lm(form_model_srl3pm_fxd, data=df_FDHI_srl, weights=reg_wt)
#residuals
df_FDHI_srl$srl1m_fxd_res  <- resid(fit_model_srl1m_fxd)
df_FDHI_srl$srl3pm_fxd_res <- resid(fit_model_srl3pm_fxd)

#coefficients
df_model_pref_srl_mag_coef <-                                       as.data.frame(t(coef(fit_model_srl1m_fxd)),  row.names='model_srl1_fxd (preferred)')
df_model_pref_srl_mag_coef <- bind_rows(df_model_pref_srl_mag_coef, as.data.frame(t(coef(fit_model_srl3pm_fxd)), row.names='model_srl3p_fxd (alternative)'))
df_model_pref_srl_mag_coef['model_srl1_fxd (preferred)',   'x1']      <- 1
df_model_pref_srl_mag_coef['model_srl3p_fxd (alternative)','x3prime'] <- 1
df_model_pref_srl_mag_coef['model_srl1_fxd (preferred)',   'sigma']   <- sigma(fit_model_srl1m_fxd)
df_model_pref_srl_mag_coef['model_srl3p_fxd (alternative)','sigma']   <- sigma(fit_model_srl3pm_fxd)

#rename columns
colnames(df_model_pref_srl_mag_coef)[colnames(df_model_pref_srl_mag_coef)=='(Intercept)'] <- 'intrcp'
colnames(df_model_pref_srl_mag_coef)[colnames(df_model_pref_srl_mag_coef)=='x1']          <- 'X1'
colnames(df_model_pref_srl_mag_coef)[colnames(df_model_pref_srl_mag_coef)=='x3prime']     <- 'X3prime'
#reorder model coefficients
df_model_pref_srl_mag_coef <- df_model_pref_srl_mag_coef[,c('intrcp','X1','X3prime','FD','sigma')]


## Comparison of different models
fprintf("SRL Modeling:
        \tModel1 fxd. (preferred):\n\t\tLogLik: %.2f\n\t\tAIC: %.2f\n\n
        \tModel3' fxd. (alternative):\n\t\tLogLik: %.2f\n\t\tAIC: %.2f\n\n",
        logLik(fit_model_srl1m_fxd), AIC(fit_model_srl1m_fxd), logLik(fit_model_srl3pm_fxd), AIC(fit_model_srl3pm_fxd) )

## SRL Modeling:
##          Model1 fxd. (preferred):
##      LogLik: 14.49
##      AIC: -22.99
## 
## 
##          Model3' fxd. (alternative):
##      LogLik: 12.05
##      AIC: -18.10

#empirical sigma below and above width threshold of model 1
sig_m1_fxd_wu  <- sqrt(mean(df_FDHI_srl[!df_FDHI_srl$flag_wlim,'srl1m_fxd_res']^2))
sig_m1_fxd_wl  <- sqrt(mean(df_FDHI_srl[ df_FDHI_srl$flag_wlim,'srl1m_fxd_res']^2))
sig_m3p_fxd_wu <- sqrt(mean(df_FDHI_srl[!df_FDHI_srl$flag_wlim,'srl3pm_fxd_res']^2))
sig_m3p_fxd_wl <- sqrt(mean(df_FDHI_srl[ df_FDHI_srl$flag_wlim,'srl3pm_fxd_res']^2))
fprintf("Standard Deviations:\n\tModel 1 fxd. (preferred):\n\t\tEmpirical ratio for std of width limted to unbounded events is: %.2f\n
         Standard Deviations:\n\tModel 3' fxd. (alternative):\n\t\tEmpirical ratio for std of width limted to unbounded events is: %.2f\n\n",
        sig_m1_fxd_wu/sig_m1_fxd_wl, sig_m3p_fxd_wu/sig_m3p_fxd_wl)

## Standard Deviations:
##  Model 1 fxd. (preferred):
##      Empirical ratio for std of width limted to unbounded events is: 1.92
## 
##          Standard Deviations:
##  Model 3' fxd. (alternative):
##      Empirical ratio for std of width limted to unbounded events is: 1.77

Table 6. SRL Preferred Model Coefficients

	Intercept	\(X_1\) scaling	\(X_3'\) scaling	\(F_D\)	\(\sigma\)
model_srl1_fxd (preferred)	-4.172824	1	NA	-0.1286700	0.2502627
model_srl3p_fxd (alternative)	-4.111267	NA	1	-0.1855579	0.2553302

Regression summary of model 1_fxd:

## 
## Call:
## lm(formula = form_model_srl1m_fxd, data = df_FDHI_srl, weights = reg_wt)
## 
## Weighted Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.63369 -0.17869  0.02732  0.16560  0.56149 
## 
## Coefficients:
##             Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) -4.17282    0.02429 -171.761  < 2e-16 ***
## FD          -0.12867    0.04253   -3.025  0.00304 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2503 on 120 degrees of freedom
## Multiple R-squared:  0.07086,    Adjusted R-squared:  0.06312 
## F-statistic: 9.152 on 1 and 120 DF,  p-value: 0.00304

Regression summary of model 3':

## 
## Call:
## lm(formula = form_model_srl3pm_fxd, data = df_FDHI_srl, weights = reg_wt)
## 
## Weighted Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.71561 -0.17854  0.01277  0.15730  0.55563 
## 
## Coefficients:
##             Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) -4.11127    0.02479 -165.869  < 2e-16 ***
## FD          -0.18556    0.04339   -4.276 3.84e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2553 on 120 degrees of freedom
## Multiple R-squared:  0.1322, Adjusted R-squared:  0.125 
## F-statistic: 18.29 on 1 and 120 DF,  p-value: 3.838e-05

Figure 6 presents the predictions of all candidate \(SRL\) models for Strike-slip events with a \(15 km\) and \(20 km\) width of seismogenic zone. Figure 7 presents the predictions for model 1 and model 1 fxd, Figure 8 presents the predictions for model 2 and model 2', and Figure 9 presents the predictions for model 3, model 3', and model 3' fxd.

Figure 6. Prediction of candidate SRL Models for Strike-slip events with a 15km and 20km width of seismogenic zone

Figure 7. Prediction of SRL model 0 for Strike-slip events

Figure 8. Prediction of SRL model 1 for Strike-slip events with a 15km and 20km width of seismogenic zone

Figure 9. Prediction of SRL model 2 for Strike-slip events with a 15km and 20km width of seismogenic zone

Figure 10. Prediction of SRL model 3 for Strike-slip events with a 15km and 20km width of seismogenic zone

Figures 11 and 12 present the predictions of model 1 fxd (preferred model) and model 3' fxd (alternative model) for Strike-slip events with a \(15 km\) and \(20 km\) width of seismogenic zone, respectively.

Figure 11. Prediction of preferred SRL Model for Strike-slip events with a 15km and 20km width of seismogenic zone

Figure 12. Prediction of alternative SRL Model for Strike-slip events with a 15km and 20km width of seismogenic zone

Figures 13 and 14 present the predictions of model 1 fxd (preferred model) and model 3' fxd (alternative model) for Strike-slip and Dip-slip events with a \(15 km\) width of seismogenic zone.

Figure 13. Prediction of preferred SRL Model for Strike-slip and Dip-slip events and 15km width of seismogenic zone

Figure 14. Prediction of alternative SRL Model for Strike-slip and Dip-slip events and 15km width of seismogenic zone

Figure 15 compares the predictions of model 0, model 1 fxd, and model' 3 fxd for Strike-slip events with a \(15 km\) and \(20 km\) width of seismogenic zone. All models have similar scaling for unbounded ruptures, but model 1 fxd and model' 3 fxd also capture the change in scaling for width-limited ruptures leading to an overall better predictive performance.

Figure 15. Prediction of Candidate SRL Models for Strike-slip events with a 15km and 20km width of seismogenic zone. Comparison with unbounded and width-limited events

Figure 16 shows the model 0 regression residuals versus magnitude.

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 16. Linear model (Model 0, WC type) Residuals versus Magnitude

Figure 17 shows the regression residuals versus magnitude of model 1 fxd (preferred model).

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 17. Prefered Model (Model 1, D~sqrt(A), fixed slope) Residuals versus Magnitude

Figure 18 shows the regression residuals versus magnitude of model 3' fxd (alternative model).

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 18. Alternative Model (Model 3’, D~sqrt(A) - L, fixed slope) Residuals versus Magnitude

Figure 19 to 21 show regression residuals versus magnitude for model 1, model 2, and model 3.

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 19. Candidate Model (Model 1, D~sqrt(A)) Residuals versus Magnitude

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 20. Candidate Model (Model 2, D~W) Residuals versus Magnitude

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 21. Candidate Model (Model 3, D~L) Residuals versus Magnitude

Figure 22 and 23 show regression residuals versus magnitude for model 2' and model 3'.

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 22. Candidate Model (Model 2’, D~sqrt(A)-L) Residuals versus Magnitude

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 23. Candidate Model (Model 3’, D~sqrt(A)-L) Residuals versus Magnitude

A rupture-width transition parameter \(\delta\) is defined as \(\delta = \log_{10}(W_{unb}) - \log_{10}(W_{lim})\); \(W_{unb}\) is the theoretical rupture width for an infinite seismogenic zone estimated from the first leg of model w1 (\(\log_{10}(W_{unb}) = \beta_1 \mathbf{M} + \beta_2\)). A negative value of \(\delta\) corresponds to a unbounded rupture, whereas a positive corresponds to a width-limited rupture. Figure 24 and 25 show the regression residuals versus \(\delta\) for the preferred (model 1 fxd) and alternative (model 3' fxd) models. The solid black lines correspond to the rupture-width transition dependent standard deviation.

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 24 Prefered Model (Model 1, D~sqrt(A), fixed slope) Residuals versus Width transition

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).

Figure 25 Alternative Model (Model 3’, D~sqrt(A) - L, fixed slope) Residuals versus Width transition

## Warning: Removed 1 rows containing missing values (`geom_pointrange()`).
## Removed 1 rows containing missing values (`geom_pointrange()`).
## Removed 1 rows containing missing values (`geom_pointrange()`).
## Removed 1 rows containing missing values (`geom_pointrange()`).
## Removed 1 rows containing missing values (`geom_pointrange()`).

Comparison With Existing Models

Figure 23 compares the preferred model with existing \(SLR\) relationships. WC94 All and WC94 SS correspond to the Wells and Coppersmith (1994) \(SRL ~ \mathbf{M}\) scaling relationships for All types of slip and Strike-slip events. W08 All, W08 SS, and W08 R correspond to the Wesnousky (2008) \(SRL ~ \mathbf{M}\) scaling relationships for All, Dip-slip, Strike-slip, and Reverse Shallow-Crustal events. L10 DS, L10 SS, and L10 SRC correspond to the Leonard (2010) \(SRL ~ \mathbf{M}\) scaling relationships for Dip-slip, Strike-slip, and Stable-Continental events. L14 Interpalte SS and L14 Interpalte DS correspond to the Leonard (2014) \(SRL ~ \mathbf{M}\) scaling relationships for Strike-slip and Dip-slip Interplate events. L14 Interpalte SS and L14 Interpalte DS correspond to the Leonard (2014) \(SRL ~ \mathbf{M}\) scaling relationships for Strike-slip and Dip-slip Interplate events, while L14 SCR DS corresponds to Dip-slip Stable-Continental events. WY15 OLS and WY15 Deming correspond to the Wells and Youngs (2015) \(SRL ~ \mathbf{M}\) scaling relationships using ordinary-least-squares and errors-in-variables regression models. The proposed model is in agreement with the existing models over different magnitude ranges. It is in good agreement with the Leonard (2010) scaling relationship for small to moderate size events, and in good agreement with the Wells and Coppersmith (1994) and Wells and Youngs (2015) scaling relationships at large size events. A potential reason for the agreement at different magnitude ranges is that the dataset used by Leonard (2010) was primarily composed of events ranging from \(\mathbf{M}~5\) to \(7\), while datasets used by Wells and Coppersmith (1994) and Wells and Youngs (2015) was primarily composed of events ranging from \(\mathbf{M}~6\) to \(7.8\).

## Warning: Removed 13 rows containing missing values (`geom_line()`).

## Warning: Removed 26 rows containing missing values (`geom_line()`).

## Warning: Removed 6 rows containing missing values (`geom_line()`).

## Warning: Removed 16 rows containing missing values (`geom_line()`).

## Warning: Removed 3 rows containing missing values (`geom_line()`).

## Warning: Removed 4 rows containing missing values (`geom_line()`).

## Warning: Removed 13 rows containing missing values (`geom_line()`).

## Warning: Removed 26 rows containing missing values (`geom_line()`).

## Warning: Removed 6 rows containing missing values (`geom_line()`).

## Warning: Removed 16 rows containing missing values (`geom_line()`).

## Warning: Removed 3 rows containing missing values (`geom_line()`).

Figure 26. Comparison Selected Models (Model 1, D~sqrt(A), fixed slope) & (Model 3’, D~sqrt(A) - L, fixed slope) with existing Active Shallow Crustal Strike-Slip SLR relationships

Figure 27. Comparison Selected Models (Model 1, D~sqrt(A), fixed slope) & (Model 3’, D~sqrt(A) - L, fixed slope) with existing Active Shallow Crustal Dip-Slip SLR relationships

## Warning: Removed 4 rows containing missing values (`geom_line()`).

Figure 28. Comparison Selected Models (Model 1, D~sqrt(A), fixed slope) & (Model 3’, D~sqrt(A) - L, fixed slope) with existing Stable Continental Region SLR relationships

## Warning: Removed 13 rows containing missing values (`geom_line()`).

## Warning: Removed 26 rows containing missing values (`geom_line()`).

## Warning: Removed 60 rows containing missing values (`geom_line()`).

## Warning: Removed 4 rows containing missing values (`geom_line()`).

## Warning: Removed 6 rows containing missing values (`geom_line()`).

## Warning: Removed 16 rows containing missing values (`geom_line()`).

## Warning: Removed 3 rows containing missing values (`geom_line()`).

Figure 29. Comparison Selected Models (Model 1, D~sqrt(A), fixed slope) & (Model 3’, D~sqrt(A) - L, fixed slope) with existing SLR relationships

Acknowledgements

• Alexandra Sarmiento (discussions & comments)
• Stephane Baize (data)

References

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